Neutron Scattering from Magnetic Materials part 4 pdf

84 T. Chatterji Fig. 44. Crystal structures of Ruddlesden–Popper phases. determined by Ruddlesden and Popper in 1958 [113]. The Rietveld profile refinements on neutron and (both laboratory and synchrotron) X-ray powder diffraction data produced a huge amount of literature on the structure analysis of Ruddlesden–Popper phases. As al- ready mentioned in the previous section the structure is flexible with respect to the cation order disorder between the 12-coordinated perovskite (P) and 9-coordinated rock-salt (R) sites and also with respect to the octahedral distortion. To make the situation more compli- cated there exists the possibility of phase separation mentioned before. The possibility of electronic phase separation also exists. Despite these complications accurate structural pa- rameters have been determinedboth as a function of hole doping x and also as a function of temperature. Also available are structural parameters as a function of both hydrostatic pres- sure and applied magnetic field. Of special interest are the Mn–O bond lengths. There is only one Mn site (4e :00z), but there are three O sites in the structure, O(1)(2a :000), O(2) (4e :00z) and O(3)(8g :0 1 2 z). As mentioned before there are two cation sites: La/Sr(P) (2b :00 1 2 ) and La/Sr(R) (4e :00z). The essential result of the structural evolution at room temperature with x is that the JT distortion ∆ JT which is about 1.04 for x = 0.3 decreases continuously and becomes very small, ∆ JT = 1.005 at x = 0.5 [114]. This result remains qualitatively the same at low temperature, but ∆ JT is slightly smaller [114]. The tempera- ture variation of ∆ JT for La 1.2 Sr 1.8 Mn 2 O 7 (x = 0.4) shows a minimum close to T C [115]. Investigation of the crystal structure of the same compound [115] as the temperature is lowered through T C in a field of 0.6 T reveals a significant magnetostriction. The equa- torial Mn–O bond contracts and the apical Mn–O bond expands in a magnetic field. For Magnetic structures 85 LaSr 2 Mn 2 O 7 (x = 0.5) charge-orbital ordering [116] takes place at T CO ≈ 225 K. The intensity of the superlattice reflections corresponding to the charge-orbital ordering in- creases continuously below T CO ≈ 225 K. At about 165 K it shows a maximum and then starts decreasing and becomes very small at about 100 K. The intensity of the superlat- tice reflection starts increasing again below 50 K. Ling et al. [117] have investigated the crystal and magnetic structure in the doping range 0.5  x  1.0. The crystal structure be- comes orthorhombic (Immm)0.8 <x<1.0 but the transformation is never complete. The orthorhombic phase coexists with the tetragonal phase. The magnetic structure of La 1−2x Sr 1+2x Mn 2 O 7 has been investigated for 0.3  x  1.0 by various authors [117–121]. Essentially there are three types of phases: ferromagnetic (FM) and antiferromagnetic (AFM) and canted (C) phases. There are two types of fer- romagnetic phases FM-I and FM-II. In FM-I the magnetic moments of the Mn ions are in the a–b plane whereas in FM-II the moments are parallel to the c axis. There are two essential types of antiferromagnetic phases. For LaSr 2 Mn 2 O 7 (x = 0.5) the antiferromag- netic phase (AFM-I) is similar to the A-type AFM phase observed in perovskite. Here the individual ferromagnetic layers of the bilayers are antiferromagnetically stacked. The bi- layers as units are also antiferromagnetically stacked. In the other type of AFM-II phase, the two individual ferromagnetic layers of the bilayers are ferromagnetically stacked, but the bilayers as units are stacked antiferromagnetically. There exist a third type of phase called the canted phase. This is similar to the AFM-I phase except that the individual lay- ers of the bilayers are canted by an angle which is different from 180 degrees. The canted phase can be thought of a combination of FM-I and AFM-I phases in which there exist both ferromagnetic and antiferromagnetic components. For x = 0.3–0.4 ferromagnetism dominates at lower temperature whereas for x>0.4 antiferromagnetism dominates. The saturated magnetic moment is about 3µ B at low temperatures. Ling et al. [117] have pub- lished a phase diagram for both crystal and magnetic phases in the concentration range 0.3 <x<1.0. This phase diagram shows the type-C AF phase in the concentration range 0.74 <x<0.9 and the G-type AF phase in the range 0.915 <x<1.0. Close to x = 0.9 there is a small region where C- and G-type AF phases coexist. There is no long-range magnetic order in the range 0.64 <x<0.74. Recently Okamoto et al. [122] have investi- gated the correlation between the orbital structure and the magnetic ordering temperature and the type of magnetic structure of the bilayer manganites at low temperatures. The two e g orbitals 3d 3z 2 −r 2 and 3d x 2 −y 2 inaMn 3+ ion are split in the crystal field of the layered structure and only one of them is occupied by the e g electron. The occupied orbital con- trols the anisotropy of the magnetic interaction and also its strength. The bilayer manganite La 1−2x Sr 1+2x Mn 2 O 7 becomes a metallic ferromagnet for a hole doping 0.32  x  0.42. The Curie temperature T C is about 110 K for x = 0.32. It increases with increasing x and becomes highest T C ≈ 130 K for x = 0.35. On further increasing x, T C decreases again and becomes again T C ≈110 K for about x = 0.42. The magnetic structure deviates from a simple ferromagnetic for x<0.32 and also for x>0.42. The bilayer manganite La 1−2x Sr 1+2x Mn 2 O 7 for x = 0.4 is a quasi-two-dimensional fer- romagnet which shows colossal magnetoresistive properties and has therefore been inves- tigated very quite intensively. In Chapter 6 we will describe the spin dynamics of this quasi-2D ferromagnet in detail. The ferromagnetic La 1.2 Sr 1.8 Mn 2 O 7 has Curie temper- ature T C ≈ 128 K. The ferromagnetic phase transition was determined by neutron dif- 86 T. Chatterji Fig. 45. (a) Temperature variation of the magnetic contribution of the intensity of the 110 reflection of La 1.2 Sr 1.8 Mn 2 O 7 . (b) Temperature variation of the total intensity of the 110 Bragg reflection close to T C .The continuous curve is a power-law fit of the data to (13). (c) Temperature variation of the integrated intensity of the rod scattering by doing a Q scan perpendicular to the rod through the reciprocal point Q =(0, 1, 1.833). Magnetic structures 87 fraction. Figure 45(a) shows the temperature variation of the magnetic contribution of the intensity of the 110 reflection. The magnetic intensity decreases continuously with increas- ing temperature and becomes zero at T C ≈128 K. Figure 45(b) shows the temperature vari- ation of the total intensity of the 110 Bragg reflection close to T C and a fit to the equation I = I n +I 0  T C −T T C  2β , (13) where I n is the nuclear contribution to the intensity, I 0 is the magnetic intensity at T = 0, T C is the ferromagnetic transition or Curie temperature and β is the critical exponent. The least squares fit gave T C = 128.7 ±0.1 K and β = 0.35 ±0.01. The critical exponent ob- tained from the fit is closer to the three-dimensional (3D) Heisenberg value, β =0.38, than to the two-dimensional (2D) Ising value, β = 0.125. Although La 1.2 Sr 1.8 Mn 2 O 7 behaves like a quasi-2D ferromagnet, the ferromagnetic phase transition at T C ≈ 128 K caused by relatively weak inter-bilayer exchange interaction is ultimately of 3D-Heisenberg type. A second determination of T C from the temperature variation of the integrated intensity of the rod scattering by doing a Q scan perpendicular to the rod through the reciprocal lattice point Q =(0, 1, 1.833) (Figure 45(c)) also gave T C ≈128 K. The spin dynamics of the quasi-2D bilayer manganite La 1.2 Sr 1.8 Mn 2 O 7 has been inves- tigated in detail by inelastic and quasielastic neutron scattering, which will be described in Chapters 6 and 7. 7. Concluding remarks In the present chapter we have described some of the magnetic structures that have been determined by neutron diffraction during the past half a century. The intricate arrangement of spins in magnetic solids at low temperatures have fascinated researchers in the field of magnetism ever since the pioneering work of Shull and Smart [3] allowed them to have a glimpse in this strange world. Giving a few hints for solving magnetic structures from polycrystalline sample or from single crystals, we have described the most frequently en- countered spin arrangements in high symmetry magnetic solids. Although these structures appear simple, one must note the difficulty arising from multi-k ordering. The multi-k structures cannot be distinguished from the single-k structure by neutron diffraction unless one applies magnetic fields and uniaxial stresses. We then introduced the more complex magnetic structures found in rare-earth elements and other magnetic solids. We described in some details the incommensurate magnetic structures which appear below the magnetic ordering temperature and the interesting phase transitions they undergo at lower tempera- tures. We also describedand discussedmagnetic phasetransitions caused by the application of magnetic field and pressure. Qualitative and phenomenological arguments are given in some cases to rationalize such structures. The magnetic structures of important electronic materials, namely, the high temperature superconducting cuprates and colossal magnetore- sistive manganites, have also been considered. We have, however, left out many important magnetic structures that exist in heavy fermion and other actinide compounds. There exist 88 T. Chatterji numerous review articles on heavy fermion and actinide compounds and interested read- ers are advised to consult those. 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CHAPTER 3 Representation Analysis of Magnetic Structures Rafik Ballou Laboratoire Louis Néel, CNRS, B.P. 166, 38042 Grenoble cedex 9, France E-mail: ballou@grenoble.cnrs.fr Bachir Ouladdiaf Institut Laue–Langevin, B.P. 156, 38042 Grenoble cedex 9, France E-mail: ouladdiaf@ill.fr Contents Introduction . . . . . . . . . . . . . 95 1.Crystallographicpreliminaries 96 2.Mathematicsofrepresentations 101 2.1.Matrixrepresentations 105 2.2. Irreducible matrix representations of space groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.3.Matrixcorepresentations 116 2.4. Irreducible matrix corepresentations of magnetic space groups . . . . . . . . . . . . . . . . . . . . 121 3.Analysisofmagneticstructures 124 3.1. Analysis without time inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.2.Analysiswithtimeinversion 130 4.Practicalworkingscheme 133 5.Application 135 5.1. First example,  k =0 135 5.2. Second example,  k =0 146 References 150 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited by Tapan Chatterji © 2006 Elsevier B.V. All rights reserved 93 [...]... analysis of magnetic structures 95 Introduction Neutron diffraction is the most powerful experimental technique for the determination of the magnetic structures of crystals: as from the directions of the magnetic scatterings with respect to the incident neutron beam, the magnetic periodicity (defined by wave vectors k in the reciprocal space) can be deduced, and from the intensities of these scatterings,... decompositions The coefficients of these combinations are the unknown to be calculated from the magnetic contributions to the neutron intensities to deduce the actual magnetic structure of the crystal A further simplification arises when the magnetic structure is stabilized through a second-order phase transition from the paramagnetic state for it would then be fully described by a single irreducible matrix... AG A is any of the elements of M − G, but once chosen must be kept fixed Using this coset expansion, all the magnetic space groups M can be deduced from the 230 space groups G 142 1 magnetic space groups are thus obtained [17,29–31] 230 of them are grey-space groups, expanding as M = G + ΘG 6 74 others are black-and-white space groups with an ordinary Bravais lattice, expanding as M = H +Θ(G−H ), where... orientations of the magnetic moments can be computed Using the trial-and-error method to solve for the orientations, however, becomes tedious and inefficient if the number of the magnetic moments in the primitive cell of the crystal structure is large Complex magnetic structures can be solved by using methods of the mathematical theory of representations of groups to select solely the magnetic configurations... were concerned with matrix corepresentations Concrete instances of magnetic structures deduced from neutron patterns with the help of matrix representations (or of matrix corepresentations) can also be found in the literature, of which we mention a few [11–15] in our reference list, but these are more exceptional than systematic A lot of magnetic structures are published without any indication of a use... group Gk (or of the magnetic space group Mk ) as defined below, on the k-component moments Su (k) (u ∈ s) of each magnetic site s in the crystal Next, these matrix representations (or matrix corepresentations) are reduced over the irreducible matrix representations of the space group Gk (or the irreducible matrix corepresentations of the magnetic space group Mk ) The symmetry allowed magnetic structures... from its action on the different vectors em of the chosen basis {en } in E as ˆ ˆ R ρ (g)(em ) = n Λρ (g)nm en , where the d 2 coefficients Λρ (g)nm are the elements of a ˆ ˆ d × d matrix representative Λρ (g) of R ρ (g) As from the homomorphism rule for R we deduce that Λσρ (hg) = Λσ (h)σ Λρ (g) ∀h ∈ G, ∀g ∈ G, ( 14) where σ {Λρ (g)} is the matrix the elements of which are σ (Λρ (g)nm ) Equation ( 14) ... reciprocal space, is the one deduced from the position of the magnetic reflections in the neutron patterns Since Km · Rn = 2π × (integer) for any reciprocal lattice vector Km , it suffices that the wave vector k in (33) runs over all the points in the (first) Brillouin zone to get all the irreducible representations of T Notice that equation (33) is deduced as a particular instance (d = 1) of the more... of index 2 and (G − H ) contains no pure translations The remaining 517 are ordinary space Representation analysis of magnetic structures 101 Fig 1 Ordinary face centered cubic Bravais lattice (F ) and black-and-white face centered cubic Bravais lattice (FS ) generated from F with the magnetic translation t0 = (a1 + a2 + a3 )/2 groups with black-and-white Bravais lattice, expanding as M = G + Θ{ε|t0... shown in Figure 1 Any magnetic space group M contains an invariant subgroup TS of lattice translations, which can be either ordinary TS = T or black-and-white TS = T + Θ{ε|t0 }T 22 blackand-white Bravais lattices can be built up in addition to the 14 ordinary Bravais lattices The factor groups M TS are isomorphous to the isogonal point groups G0 when TS is black-and-white and to the magnetic point groups . 127 3.2.Analysiswithtimeinversion 130 4. Practicalworkingscheme 133 5.Application 135 5.1. First example,  k =0 135 5.2. Second example,  k =0 146 References 150 NEUTRON SCATTERING FROM MAGNETIC MATERIALS Edited. for about x = 0 .42 . The magnetic structure deviates from a simple ferromagnetic for x<0.32 and also for x>0 .42 . The bilayer manganite La 1−2x Sr 1+2x Mn 2 O 7 for x = 0 .4 is a quasi-two-dimensional. as from the directions of the magnetic scatterings with respect to the incident neutron beam, the magnetic periodicity (defined by wave vectors  k in the reciprocal space) can be deduced, and from